Rotational Frequency [figure skater]

[otonh210]

Homework Statement

A 45kg figure skater is spinning on the toes of her skates at 1.0 rev/s. Her arms are outstretched as far as they will go/ In this orientation, the skater can be modeled as a cylindrical torso (40kg, 20cm average diameter, 160cm tall) plus two rod-like arms (2.5kg each, 66cm long) attached to the outside of the torso. The skater then raises her arms straight above her head, where she appears to be a 45kg, 20cm diameter, 200cm tall cylinder. What is her new rotation frequency, in revolutions per second?

Homework Equations

L=Iw
Li=Lf
Ii = Ibody + 2 Iarm

The Attempt at a Solution

I did conservation of L
(Ibody + 2 Iarm)(1 rev/s) = Iwholebody (wf)
Ibody = 1/2 Mr^2 = 1/2(40kg)(.1m)^2
Iarm = 1/3Mr^2 (?)
Iwholebody = 1/2Mr^2 = 1/2(45kg)(.1m)^2

When I did it this way, I got ~ 5.7 rev/s
The answer I am supposed to be getting is 3.5 rev/s
All the resources I am using to help me figure this out are not offering any new help.
Am I supposed to be using her height? Where?

Thank you very much.

 

[Doc Al]

I did conservation of L
(Ibody + 2 Iarm)(1 rev/s) = Iwholebody (wf)
Ibody = 1/2 Mr^2 = 1/2(40kg)(.1m)^2

Good.

Iarm = 1/3Mr^2 (?)

This is the tricky part. The arms attach to the outside of the torso, not to the axis of rotation. You'll need the parallel axis theorem.

Iwholebody = 1/2Mr^2 = 1/2(45kg)(.1m)^2

Good.

Show the details of your calculations.

 

[otonh210]

I'm unsure of how to apply the Parallel Axis Theorem, because I've never heard of it until now.
Do I just add the distance from the outside of the torso [attachment point] to the axis of rotation?

Iarm = 1/3(m)(l+r)^2?
Iarm = 1/3(2.5kg)(.66m+.1m)^2
Iarm = 0.4813
Ibody = 1/2(40kg)(.1m^2)
Ibody = 0.2

Iwholebody = 1/2(45kg)(.1m^2)
Iwholebody = 0.225It was mentioned to me that I should pretend there is only one arm?
Also, that I need to find the diameter of the arms?
Do I need the height in there? Or is that extraneous information?

Thank you for the reply!

 

[otonh210]

I tried the math above, and it does not work.

Do I need to use a different equation for the arms?
Ibody+MD^2 (?)

If so, what do I use for D? The distance from the edge of the torso to the axis of rotation? The distance from the centre of mass of the arm to the axis of rotation? The distance from the far end of the arm to the axis of rotation?

 

[Doc Al]

I'm unsure of how to apply the Parallel Axis Theorem, because I've never heard of it until now.
Do I just add the distance from the outside of the torso [attachment point] to the axis of rotation?

Iarm = 1/3(m)(l+r)^2?
Iarm = 1/3(2.5kg)(.66m+.1m)^2
Iarm = 0.4813

No, that would be true if the arms were 0.76 long and were directly attached to the axis.

Ibody = 1/2(40kg)(.1m^2)
Ibody = 0.2

Iwholebody = 1/2(45kg)(.1m^2)
Iwholebody = 0.225

OK.

It was mentioned to me that I should pretend there is only one arm?

That's OK. (As long as that one arm has a mass of 5 kg.)

Also, that I need to find the diameter of the arms?

Nah.

Do I need the height in there? Or is that extraneous information?

You won't need it. (Note that the formula for rotational inertia of a cylinder does not contain the height.)

What's the initial rotational rate? 1 rev/sec? (You had written 1/0, a typo I presume.)

 

[otonh210]

Is Parallel Axis Theorem Ic + MD^2?
I tried the above, with many different Ds, but none seemed to give me the right answer.

Yes, that is a typo. It is supposed to be 1.0 rev/s

Each arm has a mass of 2.5 kg. The person mentioned to me that they got the correct answer by pretending it was one giant arm of 5kg.

 

[Doc Al]

Is Parallel Axis Theorem Ic + MD^2?

Yes.

I tried the above, with many different Ds, but none seemed to give me the right answer.

I don't see how you can get the 3.5 rev/s either. Why do you think it's correct? (If this is a textbook problem, tell me the textbook.)

Yes, that is a typo. It is supposed to be 1.0 rev/s

OK.

Each arm has a mass of 2.5 kg. The person mentioned to me that they got the correct answer by pretending it was one giant arm of 5kg.

That's fine. Since the arms have the same shape, the rotational inertia of both is just twice that of one.

 

[otonh210]

For my Physics class, we have sets of problems that we must hand in each week. They tell us the answers, so that we know we are doing it correctly [but not how to get there].

The answer they gave us for this problem was 3.5 rev/s. It appears to be 12.90, but I am unsure of which textbook. Possibly 'Mastering Physics' or 'Physics for Scientists and Engineers'.


For Parallel Axis Theorem, is the Ic just the Iarm [1/3Mr^2] and then I have to add MD^2, where D is the distance from the edge of the torso to the axis of rotation?
And then I add two of those to the Ibody?

 

[Doc Al]

Possibly 'Mastering Physics' or 'Physics for Scientists and Engineers'.

Who's the author?

For Parallel Axis Theorem, is the Ic just the Iarm [1/3Mr^2] and then I have to add MD^2,

Yes.

where D is the distance from the edge of the torso to the axis of rotation?

No, D is the distance between the center of mass and the axis.

And then I add two of those to the Ibody?

Yes.

 

[otonh210]

Ah, Mastering Physics is a program that comes with 'Physics for Scientists and Engineers' to help people study. Author for Physics for S&E is Randall D Knight, I believe.

Thank you very very much for your assistance!

I am now getting 8.22 rev/s as a final answer. Does this sound more correct to you?

 

[Doc Al]

Ah, Mastering Physics is a program that comes with 'Physics for Scientists and Engineers' to help people study. Author for Physics for S&E is Randall D Knight, I believe.

I have his textbook, but not the Mastering Physics program.

I am now getting 8.22 rev/s as a final answer. Does this sound more correct to you?

I get a different answer. (I get an answer very close to the one you first posted.) What did you get for the I of the outstretched arms?

 

[otonh210]

For Iarm I did,

Iarm = (1/3)(m)(r^2) + (m)(d^2)
Iarm = (1/3(5kg)(.66m^2) + (5kg) (.43m^2)

Iarm = 1.6505


then,
[Ibody + Iarm]/Iwholebody = wf

[0.2 + 1.6505]/0.225 = 8.22 rev/s

 

[Doc Al]

For Iarm I did,

Iarm = (1/3)(m)(r^2) + (m)(d^2)
Iarm = (1/3(5kg)(.66m^2) + (5kg) (.43m^2)

For a thin rod about its center of mass, the formula is (1/12)(m)(r^2).

 

[otonh210]

Okay. I thought you had to use the rod about one end, because they are attached to the torso which is rotating.

Thank you SO much for your help though. I appreciate it a lot!

 

[Doc Al]

I thought you had to use the rod about one end, because they are attached to the torso which is rotating.

Whenever you use the parallel axis theorem, you start with the rotational inertia about the center of mass.

 

[otonh210]

Ah. I have clearly absorbed nothing from the lessons on rotation.

So,

Iarm = 1/12 (5kg)(.66m^2) + (5kg)(.43m^2)
Iarm = 1.106


wf = 5.804 rev/s

 

[Doc Al]

So,

Iarm = 1/12 (5kg)(.66m^2) + (5kg)(.43m^2)
Iarm = 1.106


wf = 5.804 rev/s

Good. This is how I would do this problem, based on the description given.